This is an intriguing question which I wanted to ask myself. From what I could gather, there are two points here, one theoretical and one empiric.
As far as I know, caffeine is one of the fastest-extracting compounds in coffee, such that it is extracted almost immediately, so, logically, the amount of caffeine in a brew would only be determined by the amount of beans that go into it and not the brewing method. Which means that whatever comes out of the AeroPress could not possibly have that much more caffeine than other methods. Particularly, not significantly more than an espresso shot, since the amounts of coffee used are about the same.
My own experience, however, suggests otherwise.
When I first got my AeroPress, I (naturally) brewed some coffee with it. Upon drinking just half of the brew, I got a kick stronger than after anything I'd ever consumed before. Several days later I hosted a party where I served AeroPress coffee, and about two thirds of the guests independently noted that the coffee was unusually strong.
Solving this contradiction seems pretty hard. Right now I see several (not entirely exclusive) possibilities:
- first, that AeroPress brew really does have a significantly higher caffeine content.
Pretty unlikely, but the possibility is nonzero.
second, that AeroPress brew contains high concentrations of some non-caffeine compounds, which:
- are produced in significantly lower quantities with other brewing methods, and
- strongly contribute to the typical effects of consuming lots of coffee.
I don't know how likely this is, since I'm not aware of whether caffeine is the only major contributor to the effects of coffee.
finally, that it is mainly a placebo-like psychological effect.
Also unlikely, since the guests at the party probably did not have any prior preconceptions about AeroPress coffee.
Bottom line: coffee from the AeroPress does tend to be perceived as stronger than average by a variety of people (including stronger after-coffee effects), but caffeine might not be the reason.